Development of energy efficient radio frequency (RF) power amplifiers (PAs) continues to be an area of focus for both industrial and academic researchers. This interest has been motivated in part by the ever increasing high peak to average power ratio (PAPR) and bandwidth of recent communication signal standards. Innovative PA architectures for use with extended back-off and bandwidth, such as Doherty PAs and envelope tracking, have already been devised and successfully demonstrated. However, the efficiency enhancement achieved by these architectures has been compromised by the accentuation of signal distortion problems. The search for a solution to address the distortion exhibited by these RF PAs has been a prolific area of research which has led to the development of several linearization techniques; namely feed-forward, feedback and predistortion techniques.
While the predistortion technique, particularly its baseband digital form, has been widely adopted to mitigate the distortions generated by RF PAs in 3rd generation (3G) wireless base stations, current techniques do not meet the challenges of 4th generation (4G) wireless networks using wideband and multi-standard signals. Investigators of digital predistortion (DPD) have struggled with linearization capacity and implementation complexity. Over time, DPD schemes have evolved from simple memoryless models to more sophisticated types which attempt to mitigate the memory effects that gain intensity as the signal bandwidth widens.
Several DPD schemes with memory have been proposed, mainly consisting of derivations of the low pass equivalent (LPE) Volterra series. In order to address the complexity burden of the LPE Volterra series, these derivations discard kernels that are considered negligible. One popular example is the memory polynomial model, where several polynomials are applied to the delayed input signals' samples separately and delayed cross-terms are discarded. Another approach, dynamic deviation reduction (DDR), has been employed to prune the LPE Volterra series. The approach begins by reformulating an LPE Volterra series expression to assemble terms with the same dynamic distortion order. By setting a value for the maximum allowed dynamic order, such a reformulation allows a simple elimination of distortion terms of higher dynamic order in the LPE Volterra series, consequently reducing its complexity. A DDR based LPE Volterra series with a preset value for the dynamic order equal to one or two has been used to linearize various RF PAs and significant linearization results were reported for different types of signals. Yet, as the signals' bandwidth increases, e.g., inter-band and inter-band carrier aggregated signals, the PA's memory effects gain intensity, and consequently a larger value for the dynamic parameter of the DDR technique is needed and eventually the complexity of the pruned LPE Volterra series undesirably approaches the un-pruned series.
Recently, several multi-band behavioral models suitable for inter-band carrier aggregated scenarios have been reported. The models are suited for carrier aggregation scenarios when the frequency separation between the two component carriers is significantly larger than the bandwidth of the individual component carriers. Consequently, these models operate at a speed proportional to the bandwidth of the individual component carriers and not to the frequency separation. However, in the case of intra-band carrier aggregated signals, the frequency separation is significantly lower. Hence, the spectrum regrowth (about five times the bandwidth of component carrier bandwidth) engendered at the different component carriers can overlap. This spectrum overlap hinders the applicability of multi-band models since these models require the acquisition of the individual envelopes of the component carriers at the output of the PA. Hence, intra-band carrier aggregated systems need to be handled as single-input single-output systems, and call for accurate and low-complex behavioral modelling schemes.
Some have distinguished two RF PA modeling strategies. The first approach approximates the behavior of the PA using its pass band input and output signals. The inherent complexity associated with this approach has limited the pass band model's adoption in the area of RF PA modeling and linearization. Instead, the LPE modeling approach is used. It is applied to the complex envelope of the RF input/output signals and requires much simpler measurement hardware and reduced computation as compared to its pass band counterpart. In fact, the LPE modeling approach capitalizes on the band limited characteristics of communication signals, thus limiting approximation efforts to the PA distortions which affect the signal envelope around the carrier frequency.
Essentially, the PA is treated as an envelope processing system. Therefore, if the real RF PA behavior is expressed as equation (1):y(t)=ƒ(x(t)),  (1)where x(t) is the real RF input signal, as shown by PA 2, y(t) is the real RF output signal and ƒ is a describing function (modeling the RF PA behavior), then the LPE methodology, as shown in FIG. 1, consists of modeling the equivalent envelope behavior of the PA via a low pass transformation, as shown by the PA 4. This is illustrated in equation (2){tilde over (y)}(n)={tilde over (ƒ)}({tilde over (x)}(n)),  (2)where {tilde over (x)}(n) and y(n) denote the envelopes of the input and output signals, respectively, around the carrier and {tilde over (ƒ)} designates the LPE model.
Since a PA can be treated as a nonlinear dynamic system with fading memory, the Volterra series outlined in equation (3) is a suitable modeling framework to approximate its behavior and/or synthesize the corresponding predistortion module.
                              y          ⁡                      (            t            )                          =                              ∑                          p              =              1                        NL                    ⁢                                    ∫                              -                ∞                            ∞                        ⁢                                                  ⁢                          …              ⁢                                                          ⁢                                                ∫                                      -                    ∞                                    ∞                                ⁢                                                                            h                      p                                        ⁡                                          (                                                                        τ                          1                                                ,                        …                        ⁢                                                                                                  ,                                                  τ                          p                                                                    )                                                        ⁢                                                            ∏                                              j                        =                        1                                            p                                        ⁢                                                                                  ⁢                                                                  x                        ⁡                                                  (                                                      t                            -                                                          τ                              j                                                                                )                                                                    ⁢                                                                        ⅆ                                                      τ                            j                                                                          .                                                                                                                                                    (        3        )            In this model, x(t) and y(t) designate the input and output RF pass band signals respectively, and hp denotes the Volterra series' kernels. The direct application of the LPE strategy to the discrete input and output signals' envelopes yields the expression of equation (4) which is commonly used for RF PA behavioral modeling.
                              y          ⁡                      (            t            )                          =                                            ∑                              p                =                1                            NL                                      p              =                              p                +                2                                              ⁢                                    ∑                                                i                  1                                =                0                            M                        ⁢                                                  ⁢                          …              ⁢                                                          ⁢                                                ∑                                                                                    i                                                  p                          +                          1                                                                    2                                        =                                                                  i                                                  p                          -                          1                                                                    2                                                        M                                ⁢                                                      ∑                                                                                            i                                                      p                            +                            3                                                                          2                                            =                      0                                        M                                    ⁢                                                                          ⁢                                      …                    ⁢                                                                                  ⁢                                                                  ∑                                                                              i                            p                                                    =                                                      i                                                          p                              -                              1                                                                                                      M                                            ⁢                                                                                                    h                            ~                                                                                                              i                              1                                                        ,                                                                                                                  ⁢                            …                            ⁢                                                                                                                  ,                                                          i                              p                                                                                                      ·                                                                              ∏                                                          j                              =                              1                                                                                                                      (                                                                  p                                  +                                  1                                                                )                                                            /                              2                                                                                ⁢                                                                                                          ⁢                                                                                                                    x                                ~                                                            ⁡                                                              (                                                                  n                                  -                                                                      i                                    j                                                                                                  )                                                                                      ·                                                                                          ∏                                                                  j                                  =                                                                                                            (                                                                              p                                        +                                        3                                                                            )                                                                        2                                                                                                  p                                                            ⁢                                                                                                                          ⁢                                                                                                                                                                          x                                      ~                                                                        *                                                                    ⁡                                                                      (                                                                          n                                      -                                                                              i                                        j                                                                                                              )                                                                                                  .                                                                                                                                                                                                                                                                (        4        )            
In equation (4) {tilde over (x)}(n) and {tilde over (y)}(n) designate the input and output signals' envelopes around the carrier sampled at ƒs=1/τs, with n, NL, M and {tilde over (h)} representing the sample index the nonlinearity order, the memory depth and the LPE complex Volterra kernels respectively.
The LPE Volterra model of equation (4) has been used extensively in RF research. The model has been applied to develop solutions related to nonlinear communication system modeling and estimation, satellite communication, digital transmission channel equalization, multichannel nonlinear CDMA system equalization, analysis and cancellation of the inter-carrier interference in nonlinear OFDM systems, decision feedback equalization, nonlinear system and circuit analysis, data predistortion, PA modeling, and DPD. Although computationally more efficient than its pass band counterpart, the LPE Volterra series in its classical form (4) still suffers from a large number of kernels. As stated earlier, this impediment has been the key limitation to its widespread adoption for RF PA behavioral modeling and predistortion. Various attempts have been made to reduce the number of kernels required (e.g., Hammerstein, Weiner, Parallel Hammerstein, Parallel Weiner, DDR based Pruned Volterra series) but at the cost of reduced modeling accuracy.